On the ground state solution of a fractional Schrödinger equation
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- by S. Paul and Sanjiban Santra PDF
- Proc. Amer. Math. Soc. 147 (2019), 5275-5290 Request permission
Abstract:
In this paper, we discuss the asymptotic behavior of the ground state solution for a fractional Schrödinger equation with a polynomial nonlinearity in one dimension.References
- Vincenzo Ambrosio, Multiplicity of positive solutions for a class of fractional Schrödinger equations via penalization method, Ann. Mat. Pura Appl. (4) 196 (2017), no. 6, 2043–2062. MR 3714753, DOI 10.1007/s10231-017-0652-5
- Vincenzo Ambrosio, Ground states solutions for a non-linear equation involving a pseudo-relativistic Schrödinger operator, J. Math. Phys. 57 (2016), no. 5, 051502, 18. MR 3501198, DOI 10.1063/1.4949352
- Vincenzo Ambrosio, Periodic solutions for critical fractional problems, Calc. Var. Partial Differential Equations 57 (2018), no. 2, Paper No. 45, 31. MR 3766988, DOI 10.1007/s00526-018-1317-y
- R. M. Blumenthal, R. K. Getoor, and D. B. Ray, On the distribution of first hits for the symmetric stable processes, Trans. Amer. Math. Soc. 99 (1961), 540–554. MR 126885, DOI 10.1090/S0002-9947-1961-0126885-4
- K. Bogdan, T. Kulczycki, and Adam Nowak, Gradient estimates for harmonic and $q$-harmonic functions of symmetric stable processes, Illinois J. Math. 46 (2002), no. 2, 541–556. MR 1936936, DOI 10.1215/ijm/1258136210
- Juan Dávila, Manuel del Pino, and Juncheng Wei, Concentrating standing waves for the fractional nonlinear Schrödinger equation, J. Differential Equations 256 (2014), no. 2, 858–892. MR 3121716, DOI 10.1016/j.jde.2013.10.006
- Rupert L. Frank and Robert Seiringer, Non-linear ground state representations and sharp Hardy inequalities, J. Funct. Anal. 255 (2008), no. 12, 3407–3430. MR 2469027, DOI 10.1016/j.jfa.2008.05.015
- Rupert L. Frank and Enno Lenzmann, Uniqueness of non-linear ground states for fractional Laplacians in $\Bbb {R}$, Acta Math. 210 (2013), no. 2, 261–318. MR 3070568, DOI 10.1007/s11511-013-0095-9
- Nikolai Laskin, Fractional quantum mechanics and Lévy path integrals, Phys. Lett. A 268 (2000), no. 4-6, 298–305. MR 1755089, DOI 10.1016/S0375-9601(00)00201-2
- Nick Laskin, Fractional Schrödinger equation, Phys. Rev. E (3) 66 (2002), no. 5, 056108, 7. MR 1948569, DOI 10.1103/PhysRevE.66.056108
- Luca Martinazzi, Fractional Adams-Moser-Trudinger type inequalities, Nonlinear Anal. 127 (2015), 263–278. MR 3392369, DOI 10.1016/j.na.2015.06.034
- Mihai Mariş, On the existence, regularity and decay of solitary waves to a generalized Benjamin-Ono equation, Nonlinear Anal. 51 (2002), no. 6, 1073–1085. MR 1926086, DOI 10.1016/S0362-546X(01)00880-X
- René Carmona, Wen Chen Masters, and Barry Simon, Relativistic Schrödinger operators: asymptotic behavior of the eigenfunctions, J. Funct. Anal. 91 (1990), no. 1, 117–142. MR 1054115, DOI 10.1016/0022-1236(90)90049-Q
- MichałRyznar, Estimates of Green function for relativistic $\alpha$-stable process, Potential Anal. 17 (2002), no. 1, 1–23. MR 1906405, DOI 10.1023/A:1015231913916
- Eleonora Di Nezza, Giampiero Palatucci, and Enrico Valdinoci, Hitchhiker’s guide to the fractional Sobolev spaces, Bull. Sci. Math. 136 (2012), no. 5, 521–573. MR 2944369, DOI 10.1016/j.bulsci.2011.12.004
- Enea Parini and Bernhard Ruf, On the Moser-Trudinger inequality in fractional Sobolev-Slobodeckij spaces, Atti Accad. Naz. Lincei Rend. Lincei Mat. Appl. 29 (2018), no. 2, 315–319. MR 3797987, DOI 10.4171/RLM/808
- Stefano Iula, A note on the Moser-Trudinger inequality in Sobolev-Slobodeckij spaces in dimension one, Atti Accad. Naz. Lincei Rend. Lincei Mat. Appl. 28 (2017), no. 4, 871–884. MR 3729591, DOI 10.4171/RLM/789
- Xiaofeng Ren and Juncheng Wei, On a two-dimensional elliptic problem with large exponent in nonlinearity, Trans. Amer. Math. Soc. 343 (1994), no. 2, 749–763. MR 1232190, DOI 10.1090/S0002-9947-1994-1232190-7
- Xiaofeng Ren and Juncheng Wei, On a semilinear elliptic equation in $\textbf {R}^2$ when the exponent approaches infinity, J. Math. Anal. Appl. 189 (1995), no. 1, 179–193. MR 1312037, DOI 10.1006/jmaa.1995.1011
- Simone Secchi, On fractional Schrödinger equations in $\Bbb R^N$ without the Ambrosetti-Rabinowitz condition, Topol. Methods Nonlinear Anal. 47 (2016), no. 1, 19–41. MR 3469045
- Michael I. Weinstein, On the structure and formation of singularities in solutions to nonlinear dispersive evolution equations, Comm. Partial Differential Equations 11 (1986), no. 5, 545–565. MR 829596, DOI 10.1080/03605308608820435
Additional Information
- S. Paul
- Affiliation: Department of Pure Mathematics, University of Calcutta, India
- MR Author ID: 799212
- Email: santi_paul2005@yahoo.co.in
- Sanjiban Santra
- Affiliation: Centro de Investigacióne en Mathematicás, Guanajuato, Mexico
- MR Author ID: 774625
- Email: sanjibansntr385@gmail.com
- Received by editor(s): September 21, 2018
- Received by editor(s) in revised form: March 24, 2019
- Published electronically: July 8, 2019
- Communicated by: Catherine Sulem
- © Copyright 2019 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 147 (2019), 5275-5290
- MSC (2010): Primary 34A08, 34A12, 34E10
- DOI: https://doi.org/10.1090/proc/14632
- MathSciNet review: 4021087